arXiv subjects

Karl-Hermann Neeb

Publications and source records attributed to Karl-Hermann Neeb.

At least 19 recordsLinked to original sources

arXiv2026

Orthogonal pairs of Euler elements II: Geometric Bisognano--Wichmann and Spin--Statistics Theorems

Models in Algebraic Quantum Field Theory (AQFT) may be generalized including Lie groups of symmetries whose Lie algebras admit an Euler element $h$, characterized by the property that $ad h$ is diagonalizable with eigenvalues in $\{-1, 0, 1\}$. These elements becomes fundamental to the formal description of wedge localization. In this paper, we extend the geometric analysis of Euler wedges and investigate their applications within the AQFT framework. We call a pair of Euler elements $(h, k)$ orthogonal if $e^{i \pi \operatorname{ad} h}(k) = -k.$ Using the geometric framework established in our previous work, we derive both a Bisognano--Wichmann Theorem and a Spin--Statistics Theorem for nets of standard subspaces and von Neumann algebras. Our results {show} how this generalized approach recovers classical results in the AQFT literature while providing a deeper structural understanding of the underlying geometry in established models.

math-ph↗

arXiv2026

Infinite-Dimensional Lie Groups

This is a preliminary version of a book on infinite-dimensional Lie groups. It covers the basics of calculus and manifolds in the context of locally convex spaces, based on Bastiani's notion of a smooth map. Starting from this concept, we develop the basics of smooth manifolds and define Lie groups as manifolds with smooth group operations. We discuss in particular several classes of Lie groups, such as regular ones, or those with an exponential function that is a local diffeomorphism. The local theory, subgroups and quotients are explored in some detail. Classes of Lie groups that are discussed in detail include: unit groups of continuous inverse algebras, groups of smooth maps, direct limit groups and groups of diffeomorphism. We also included chapters on the topology of infinite-dimensional Lie group and on various selected topics.

math.FA↗

arXiv2026

A classification of coadjoint orbits carrying Gibbs ensembles

A coadjoint orbit $O_\lambda \subseteq {\mathfrak g}^*$ of a Lie group $G$ is said to carry a Gibbs ensemble if the set of all $x \in {\mathfrak g}$, for which the function $\alpha \mapsto e^{-\alpha(x)}$ on the orbit is integrable with respect to the Liouville measure, has non-empty interior $\Omega_\lambda$. We describe a classification of all coadjoint orbits of finite-dimensional Lie algebras with this property. In the context of Souriau's Lie group thermodynamics, the subset $\Omega_\lambda$ is the geometric temperature, a parameter space for a family of Gibbs measures on the coadjoint orbit. The corresponding Fenchel--Legendre transform maps $\Omega_\lambda/{\mathfrak z}({\mathfrak g})$ diffeomorphically onto the interior of the convex hull of the coadjoint orbit $O_\lambda$. This provides an interesting perspective on the underlying information geometry. We also show that already the integrability of $e^{-\alpha(x)}$ for one $x \in {\mathfrak g}$ implies that $\Omega_\lambda \not=\emptyset$ and that, for general Hamiltonian actions, the existence of Gibbs measures implies that the range of the momentum maps consists of coadjoint orbits $O_\lambda$ as above.

math.SG↗

arXiv2025

Nets of real subspaces on homogeneous spaces and Algebraic Quantum Field Theory

In these notes, we describe an interesting connection between unitary representations of Lie groups and nets of local algebras, as they appear in Algebraic Quantum Field Theory (AQFT). It is based on first translating the axioms for nets of operator algebras parameterized by regions in a space-time manifold into those for nets of real subspaces, and then study this structure from a perspective based on geometry and representation theory of Lie groups.

math.OA↗

arXiv2025

Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality

The current article continues our project on representation theory, Euler elements, causal homogeneous spaces and Algebraic Quantum Field Theory (AQFT). We call a pair (h,k) of Euler elements orthogonal if $e^{\pi i \ad h} k = -k$. We show that, if (h,k) and (k,h) are orthogonal, then they generate a 3-dimensional simple subalgebra. We also classify orthogonal Euler pairs in simple Lie algebras and determine the fundamental groups of adjoint Euler elements in arbitrary finite-dimensional Lie algebras. Causal complements of wedge regions in spacetimes can be related to so-called twisted complements in the space of abstract Euler wedges, defined in purely group theoretic terms. We show that any pair of twisted complements can be connected by a chain of successive complements coming from $3$-dimensional subalgebras.

math.RT↗

arXiv2025

Crowned Lie groups and nets of real subspaces

We introduce the notion of a complex crown domain for a connected Lie group $G$, and we use analytic extensions of orbit maps of antiunitary representations to these domains to construct nets of real subspaces on $G$ that are isotone, covariant and satisfy the Reeh--Schlieder and Bisognano--Wichmann conditions from Algebraic Quantum Field Theory. This provides a unifying perspective on various constructions of such nets.The representation theoretic properties of different crowns are discussed in some detail for the non-abelian $2$-dimensional Lie group ${\rm Aff}({\mathbb R})$. We also characterize the existence of nets with the above properties by a regularity condition in terms of an Euler element in the Lie algebra ${\mathfrak g}$ and show that all antiunitary representations of the split oscillator group have this property.

math.RT↗

arXiv2025

Open orbits in causal flag manifolds, modular flows and wedge regions

We study open orbits of symmetric subgroups of a simple connected Lie group G on a causal flag manifold. First we show that a flag manifold M of G carries an invariant causal structure if and only if G is hermitian of tube type and M is the conformal completion of the corresponding simple euclidean Jordan algebra, resp., the Shilov boundary of the associated symmetric tube domain. We then study open orbits in M under symmetric subgroups, also called causal Makarevic spaces, from the perspective of applications in Algebraic Quantum Field Theory (AQFT). A key motivation is the geometry of corresponding modular flows. The open orbits are reductive causal symmetric spaces, which arise in two flavors: compactly causal and non-compactly causal ones. In the non-compactly causal case we determine the corresponding Euler elements and their positivity regions. For compactly causal spaces, modular flows do not always exist and we determine when this is the case. Then the positivity regions of the modular flows are not globally hyperbolic, but these spaces contain other interesting globally hyperbolic subsets that can be described in terms of the conformally flat Jordan coordinates via Cayley charts. We discuss the Lorentzian case, involving de Sitter and anti-de Sitter space in some detail.

math.DG↗

arXiv2025

Fundamental polytope for the isometry group of an alcove

A fundamental alcove $\mathcal{A}$ is a tile in a paving of a vector space $V$ by an affine reflection group $W_{\mathrm{aff}}$. Its geometry encodes essential features of $W_{\mathrm{aff}}$, such as its affine Dynkin diagram $\widetilde{D}$ and fundamental group $Ω$. In this article we investigate its full isometry group $\mathrm{Aut}(\mathcal{A})$. It is well known that the isometry group of a regular polyhedron is generated by hyperplane reflections on its faces. Being a simplex, an alcove $\mathcal{A}$ is the simplest of polyhedra, nevertheless it is seldom a regular one. In our first main result we show that $\mathrm{Aut}(\mathcal{A})$ is isomorphic to $\mathrm{Aut}(\widetilde{D})$. Building on this connection, we establish that $\mathrm{Aut}(\mathcal{A})$ is an abstract Coxeter group, with generators given by affine isometric involutions of the ambient space. Although these involutions are seldom reflections, our second main result leverages them to construct, by slicing the Komrakov--Premet fundamental polytope $\mathcal{K}$ for the action of $Ω$, a family of fundamental polytopes for the action of $\mathrm{Aut}(\mathcal{A})$ on $\mathcal{A}$, whose vertices are contained in the vertices of $\mathcal{K}$ and whose faces are parametrized by the so-called balanced minuscule roots, which we introduce here. In an appendix, we discuss some related negative results on stratified centralizers and equivariant triangulations.

math.CO↗

arXiv2024

Elliptic domains in Lie groups

An element $g$ of a Lie group is called stably elliptic if it is contained in the interior of the set $G^e$ of elliptic elements, characterized by the property that $\mathrm{Ad}(g)$ generates a relatively compact subgroup. Stably elliptic elements appear naturally in the geometry of causal symmetric spaces and in representation theory. We characterize stably elliptic elements in terms of the fixed point algebra of $\mathrm{Ad}(g)$ and show that the connected components of the set $G^{se}$ of stably elliptic elements can be described in terms of the Weyl group action on a compactly embedded Cartan subalgebra. In the case of simple hermitian Lie groups we relate stably elliptic elements to maximal invariant cones and the associated subsemigroups. In particular we show that the basic connected component $G^{se}(0)$ can be characterized in terms of the compactness of order intervals and that $G^{se}(0)$ is globally hyperbolic with respect to the induced biinvariant causal structure.

math.DG↗

arXiv2024

On the singularities of the exponential function of a semidirect product

We show that the Fr\'echet--Lie groups of the form $C^{\infty}(M)\rtimes \mathbb{R}$ resulting from smooth flows on compact manifolds $M$ fail to be locally exponential in several cases: when at least one non-periodic orbit is locally closed, or when the flow restricts to a linear one on an orbit closure diffeomorphic to a torus. As an application, we prove that the Bondi--Metzner--Sachs group of symmetries of an asymptotically flat spacetime is not locally exponential.

math.GR↗

arXiv2024

Reflection positivity and its relation to disc, half plane and the strip

We develop a novel perspective on reflection positivity (RP) on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the Möbius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on $H^\infty$ for a suitable involution. For the strip, reflection positivity naturally connects with Kubo--Martin--Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process.

math.FA↗

arXiv2024

Covariant projective representations of Hilbert-Lie groups

Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study unitary representations of these groups from various perspectives. First, we address norm-continuous, also called bounded, representations: they are well-known for simple groups, but the general picture is more complicated. Our first main result is a characterization of the discrete decomposability of all bounded representations in terms of boundedness of the set of coroots. We also show that bounded representations of type II and III exist if the set of coroots is unbounded. Second, we use covariance with respect to a one-parameter group of automorphisms to implement some regularity. Here we develop some perturbation theory based on half Lie groups that reduces matters to the case where a ``maximal torus'' is fixed, so that compatible weight decompositions can be studied. Third, we extend the context to projective representations which are covariant for a one-parameter group of automorphisms. Here important families of representations arise from ``bounded extremal weights'', and for these the corresponding central extensions can be determined explicitly, together with all one-parameter groups for which a covariant extension exists.

math-ph↗

arXiv2024

Realization of unitary representations of the Lorentz group on de Sitter space

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups $G$ acting on a non-compactly causal symmetric space $M = G/H$, every irreducible unitary representation of $G$ can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over $M$. In the present paper we discuss this procedure for the connected Lorentz group $G = SO_{1,d}(R)_e$ acting on de Sitter space $M = dS^d$. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990's, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain $G_C^{cut}$ in the complexified group $G_C = \SO_{1,d}(C)$, which for $d = 1$ specializes to the complex cut plane $C \setminus (-\infinity, 0]$. A number of special situations is discussed specifically: (a) The case $d = 1$, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for $d > 2$ is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case $d = 3$, where we obtain rather explicit formulas for the matrix-valued spherical functions.

math-ph↗

arXiv2023

From local nets to Euler elements

Various aspects of the geometric setting of Algebraic Quantum Field Theory (AQFT) models related to representations of the Poincaré group can be studied for general Lie groups, whose Lie algebra contains an Euler element, i.e., ad h is diagonalizable with eigenvalues in {-1,0,1}. This has been explored by the authors and their collaborators during recent years. A key property in this construction is the Bisognano-Wichmann property (thermal property for wedge region algebras) concerning the geometric implementation of modular groups of local algebras. In the present paper we prove that under a natural regularity condition, geometrically implemented modular groups arising from the Bisognano-Wichmann property, are always generated by Euler elements. We also show the converse, namely that in presence of Euler elements and the Bisognano-Wichmann property, regularity and localizability hold in a quite general setting. Lastly we show that, in this generalized AQFT, in the vacuum representation, under analogous assumptions (regularity and Bisognano-Wichmann), the von Neumann algebras associated to wedge regions are type III_1 factors, a property that is well-known in the AQFT context.

math.OA↗

arXiv2023

Nets of standard subspaces on non-compactly causal symmetric spaces

Let G be a connected simple linear Lie group and H in G a symmetric subgroup such that the corresponding symmetric space G/H is non-compactly causal. We show that any irreducible unitary representation of G leads naturally to a net of standard subspaces on G/H that is isotone, covariant and has the Reeh--Schlieder and the Bisognano--Wichmann property. We also show that this result extends to the universal covering group of SL(2,R) which has some interesting application to intersections of standard subspaces associated to representations of such groups. For this a detailed study of hyperfunction and distribution vectors is needed. In particular we show that every H-finite hyperfunction vector is in fact a distribution vector.

math.RT↗

arXiv2023

Holomorphic extension of one-parameter operator groups

We study holomorphic extensions of one-parameter groups on locally convex spaces with a view to applications to KMS boundary conditions. In the first part we deal with analytic extensions of one-parameter groups of operators on locally convex spaces and in the second part we apply our results to spaces of distribution vectors of unitary representations of Lie groups. This leads to new tools that can be used to construct, from unitary Lie group representations, nets of standard subspaces, as they appear in Algebraic Quantum Field Theory. We also show that these methods fail for spaces of analytic vectors, and this in turn leads to new maximality results for domains of analytic extensions of orbit maps for unitary representations.

math.RT↗

arXiv2023

Modular geodesics and wedge domains in non-compactly causal symmetric spaces

We continue our investigation of the interplay between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory. We adopt the perspective that the geometric implementation of the modular group is given by the flow generated by an Euler element of the Lie algebra (an element defining a 3-grading). Since any Euler element of a semisimple Lie algebra specifies a canonical non-compactly causal symmetric space M = G/H, we turn in this paper to the geometry of this flow. Our main results concern the positivity region W of the flow (the corresponding wedge region): If G has trivial center, then W is connected, it coincides with the so-called observer domain, specified by a trajectory of the modular flow which at the same time is a causal geodesic, it can also be characterized in terms of a geometric KMS condition, and it has a natural structure of an equivariant fiber bundle over a Riemannian symmetric space that exhibits it as a real form of the crown domain of G/K. Among the tools that we need for these results are two observations of independent interest: a polar decomposition of the positivity domain and a convexity theorem for G-translates of open $H$-orbits in the minimal flag manifold specified by the 3-grading.

math.DG↗

arXiv2022

Wedge domains in non-compactly causal symmetric spaces

This article is part of an ongoing project aiming at the connections between causal structures on homogeneous spaces, Algebraic Quantum Field Theory (AQFT), modular theory of operator algebras and unitary representations of Lie groups. In this article we concentrate on non-compactly causal symmetric space $G/H$. This class contains the de Sitter space but also other spaces with invariant partial ordering. The central ingredient is an Euler element h in the Lie algebra of \fg. We define three different kinds of wedge domains depending on h and the causal structure on G/H. Our main result is that the connected component containing the base point eH of those seemingly different domains all agree. Furthermore we discuss the connectedness of those wedge domains. We show that each of those spaces has a natural extension to a non-compactly causal symmetric space of the form G_\C/G^c where G^c is certain real form of the complexification G_\$ of G. As G_\C/G^c is non-compactly causal it also comes with the three types of wedge domains. Our results says that the intersection of those domains with $G/H$ agrees with the wedge domains in G/H.

math-ph↗